effective equation of state for a spherically expanding pion plasma
TRANSCRIPT
PHYSICAL REVIEW D 1 JANUARY 1998VOLUME 57, NUMBER 1
Effective equation of state for a spherically expanding pion plasma
Melissa A. Lampert*
Department of Physics, University of New Hampshire, Durham, New Hampshire 03824
Carmen Molina-Parı´s†
Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545~Received 8 August 1997; published 9 December 1997!
Following a relativistic heavy ion collision, the quark-gluon plasma produced eventually undergoes a chiralphase transition. We assume that during this phase transition one can describe the dynamics of the system bythe linears model and that the expansion can be thought of as mostly radial. Because thes model is aneffective field theory there is an actual momentum cutoff~Landau pole! in the theory at around 1 GeV. Thusit is necessary to find ways of obtaining a covariantly conserved, renormalized energy-momentum tensor whenthere is a cutoff present~which breaks covariance!, in order to identify the effective equation of state of thistime evolving system. We show how to solve this technical problem and then determine the energy density andpressure of the system as a function of the proper time. We consider different initial conditions and search forinstabilities which can lead to the formation of disoriented chiral condensates. We find that the energy densityand pressure both decrease quickly, as is appropriate for a rapidly cooling system, and that the energy isnumerically conserved.@S0556-2821~98!04501-9#
PACS number~s!: 11.10.Gh, 05.70.Fh, 12.38.Mh, 25.75.2q
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I. INTRODUCTION
During a relativistic heavy ion collision, it is possible tcreate a quark-gluon plasma, which will then cool andpand, leading to hadronization. At around the same timehadronization, the system also undergoes a chiral phasesition which breaks the SU~2!3SU~2! symmetry and whichthen gives physical masses to the pion and other particThe out of equilibrium dynamics of this phase transition cbe very interesting if the expansion is fast enough so thatphase transition resembles a quench. In this case the effepion mass can go negative for short periods of proper twhich leads to a temporary exponential growth of low mmentum domains where the isospin can point in a particdirection @disoriented chiral condensate~DCC!#. This is atopic of current interest@1#, since DCCs may provide a signature of the chiral phase transition. In an earlier paper@2#the time evolution of this chiral phase transition, assuminuniform spherically symmetric expansion into the vacuuwas considered. The O~4! linear s model in the large-N ap-proximation, which incorporates both nonequilibrium aquantum effects, was studied. A range of initial conditiowas analyzed to determine which initial states could leadthe formation of DCCs. This radial expansion is interestbecause it is expected at late times in the plasma evoluand also maximizes the cooling rate and therefore the pobility of the phase transition resembling a quench.
In this paper we go beyond previous work@2# and studythe proper time evolutions of the effective hydrodynamcollective variables of the system, namely the renormalipressure and energy density. We want to see to what exprevious intuition coming from studying classical hydrod
*Electronic address: [email protected]†Electronic address: [email protected]
570556-2821/97/57~1!/83~10!/$10.00
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namical models of particle production such as Landamodel is confirmed. This requires the solution of a new tenical issue, namely how one obtains a covariantly conserenergy-momentum tensor when there is an actual cutofthe theory. As discussed in earlier work@3,13#, the linearsmodel is an effective theory with a renormalized coupliconstant of order 10, in order to agree with low energy pproperties. This leads to a maximum cutoff of around 1 Gin order to avoid problems at scales of the order of the Ldau pole. It has been shown earlier@3# that as long as themomentum cutoff determined from the sum over mode futions is below the Landau pole, there is a regime of cutowhere the continuum renormalization group flow is obeyso that one can safely say that one is in the continuumgime. For this class of problems, renormalization methobased on formal schemes such as dimensional regularizaapproaches are not very useful. Here we have a real cutophysical momentum as a result of using an effective fitheory, and we have to be very careful in order to obtaincorrect covariantly conserved energy-momentum tenwhich leads to our definitions of comoving energy densand pressure variables. The tools we use to correctly remalize the energy-momentum tensor are first to introducephysical cutoffL and then to analyze the divergences inthe components of the energy-momentum tensor usingadiabatic expansion of the mode functions. By comparour results with a covariant point-splitting approach@4#, wecan identify those terms which should survive in the covaant limit. This technique allows us to obtain finite enerdensities and pressures which then enable us to studyeffective equation of state for the evolving plasma.
In this paper, we derive the energy-momentum tensorthe O~4! linear s model, and show how to regularize anrenormalize it in order to obtain the physical energy densand pressure. We study initial conditions which have insbilities that lead to DCC formation and also stable init
83 © 1997 The American Physical Society
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84 57MELISSA A. LAMPERT AND CARMEN MOLINA-PARIS
configurations, and examine the energy density and presin both situations. We also consider the conservation ofenergy-momentum tensor as a check on our numerical mods.
The paper is organized as follows. In Sec. II we discthe model and coordinate system used, and show howconstruct the energy-momentum tensor. Then in Sec. IIIexplicitly describe the scheme used to renormalize thissor in order to perform a numerical simulation with a phycal momentum cutoff. In Sec. IV we numerically examithe conservation of energy and the equation of state ofsystem. Finally in Sec. V we discuss the results and provsome concluding remarks.
II. ENERGY-MOMENTUM TENSOR
A. Equations of motion and coordinate system
We consider the O~4! linear s model, with the classicaaction given by@2#
S@F i ,x; j i #5E d4xA2g~x!~L@F i ,x#1 j iF i !, ~2.1!
where the mesons are in a O~4! vector representation
F i5~s,pW !. ~2.2!
The Lagrangian density is
L@F i ,x#51
2gmn~x!@]mF i~x!#@]nF i~x!#2
1
2x~x!F i
2~x!
1v2
2x~x!1
1
4lx2~x!. ~2.3!
In order to quantize the system, we construct the generafunctional of the connected Green’s functions and carrythe path integral to obtain the following quantum effectiaction ~to leading order inN! @2#
G@f i ,x; j i #5S@f i ,x; j i #1iN
2Tr ln G0
21~x,x8;x!,
~2.4!
where
G021~x,x8;x!5@h1x~x!#d4~x,x8!/A2g~x!, ~2.5!
and
f i~x
iths,
.e
deor
n
tha
uc
r-
ct toantheto
a
e
dee-re
57 85EFFECTIVE EQUATION OF STATE FORA . . .
gmn5diag„1,2a2~t!,2a2~t!sinh2h,
2a2~t!sinh2h sin2u…, ~2.14!
which corresponds to a Ricci flat cosmological model wuniform expansiona(t)5t, and hyperbolic spatial sectioni.e., with curvaturek521.
It will be convenient to introduce the conformal timeugiven by
u5E t
a21~t8!dt8, ~2.15!
or equivalently
t5eu
m, ~2.16!
where we choosem5mp , the only mass scale in the systemThe metric in the coordinate system with conformal tim(u,h,u,w) has the form
gmn5C~u!diag~1,21,2sinh2h,2sinh2h sin2u!,~2.17!
with
C~u!5a2~t!5t25e2u/m2,
A2g5C2~u!sinh2h sinu. ~2.18!
In a hydrodynamical model, all the expectation valuespend only on the proper time of the system. We therefwill assume that the mean fieldsf i andx are only functionsof u, that isf i5f i(u) andx5x(u). We then write the fullquantum fieldF i in terms of its expansion about its meavalue, as follows:
F i~u,h,u,f!5f i~u!1f i~u,h,u,f!, ~2.19!
where f i are the quantum fluctuations, which include bovacuum and thermal excitations. The equations of motionthen
FC21~u!S ]2
]u2 12]
]uD1x~u!Gf i~u!5Hd i0 ,
@h1x~u!#f i~x!50, ~2.20!
where x is the four vector x5(u,h,u,w). Then forG0(x,x8;x) we find
G0~x,x8;x!5 i ^Tc$f i~x!,f i~x8!%&,
whereTc corresponds to au-ordered product, following theclosed-time-path formalism of Schwinger@6#.
In order to solve the wave equation for the quantum fltuations, we follow Parker and Fulling@7#, and write a modeexpansion forf i as follows
-e
re
-
f i~u,xW !5C21/2~u!E0
`
ds(lm
@ ai ,slm gs~u!Yslm~xW !
1a†i ,slm gs* ~u!Y slm* ~xW !#, ~2.21!
with
xW5~h,u,f!,
and
D~3!Yslm~xW !1~s211!Yslm~xW !50,
whereD (3) is the Laplacian of the three dimensional hypebolic spatial sections of curvaturek521. Then the modefunctionsgs(u) satisfy the following differential equation:
gs~u!1@s21C~u!x~u!#gs~u!50, ~2.22!
and forG0(x,x;x) we find @2#
^f i2&52 iG0~x,x;x!5C21~u!E
0
` s2ds
2p2 ~2ns11!ugs~u!u2,
~2.23!
once we have chosen a particular vector state with respewhich we shall be taking expectation values. We chooseinitial state such that the pair densities are zero, andparticle number density is finitely integrable with respectthe corresponding integration measure, namely,
^a j ,s8 l 8m8† ai ,slm&5nsd i j d~s82s!d l l 8dmm8 ,
^a j ,s8 l 8m8ai ,slm† &5~ns11!d i j d~s82s!d l l 8dmm8 ,
^a j ,s8 l 8m8† ai ,slm
† &5psd i j d~s82s!d l l 8dmm850,
^a j ,s8 l 8m8ai ,slm&5ps* d i j d~s82s!d l l 8dmm850.
Here we choose the initial particle number density to bethermal distribution,
ns5@evs~u0!/kBT21#21,
wherevs2(u0)5s21C(u0)x(u0). Notice that we can choos
the pair density to vanish (ps50), since one has the freedomto make a Bogoliubov transformation atu0 so that this al-ways remains true. The boundary conditions for the mofunctionsgs(u) is that they correspond to the positive frquency adiabatic mode functions initially. We therefochoose
gs~u0!51/A2vs~u0!,
gs~u0!52H vs~u0!
2vs~u0!1 ivs~u0!J gs~u0!. ~2.24!
The following rescalings will be useful:
f i~u!5C21/2~u!r i~u!, ~2.25a!
x~u!5C21~u!x~u!, ~2.25b!
ca
86 57MELISSA A. LAMPERT AND CARMEN MOLINA-PARIS
j i5C23/2~u! j i~u!, ~2.25c!
v5C21/2~u! v ~u!. ~2.25d!
In terms of the scaled variables, the equations of motionbe written
F d2
du2 1s21x~u!Ggs~u!50, ~2.26!
t
n
F d2
du2 211x~u!Gr i~u!5 j i~u!,
~2.27!
with the gap equation being
x~u!
l52 v 2~u!1r i
2~u!1NE0
` s2ds
2p2 ~2ns11!ugs~u!u2.
~2.28!
l state
except for
B. Construction of the stress-energy tensor
The energy-momentum tensor is defined by@8#
Tmn5~122j!~¹mF i !~¹nF i !1S 2j21
2Dgmngab~¹aF i !~¹bF i !22jF i~¹m¹nF i !11
2jgmnF ihF i
11
2~123j!xgmnF i
22gmn
xv2
22gmn
x2
4l2gmnS 12
3j
2 D j iF i . ~2.29!
Equivalently, we can write this equation as@9#
Tmn5~¹mF i !~¹nF i !1j~gmnh2¹m¹n!F i22gmn~L@F i ,x#1 j iF i !, ~2.30!
whereL@F i ,x# is given by Eq.~2.3!. We take the expectation value of the energy-momentum tensor in the thermal initiachosen, and make use of the mode expansion given in Eq.~2.21! to examine the components ofTmn .
After some algebra, we find for theTuu component
C~u!^Tuu&51
2@ r i
21~ x11212j!r i2#1~6j21!r i r i2 j ir i2
x v 2
22
x 2
4l
1N
2 E0
` s2ds
2p2 ~2ns11!$ugsu21~s21x12212j!ugsu21~6j21!~gsgs* 1gs* gs!%, ~2.31!
and similarly for theThh component
C~u!^Thh&51
2@ r i
2~124j!1~12x28j14jx !r i2#1~6j21!r i r i1~122j! j ir i1
x v2
21
x2
4l1
N
2 E0
` s2ds
2p2 ~2ns11!
3H ~124j!ugsu21F2s2
32x1
2
314j~s21x21!G ugsu21~6j21!~gsgs* 1gs* gs!J . ~2.32!
It can be easily shown that the energy-momentum tensor is diagonal and that the spatial components are all equal,a geometrical factor~see Appendix B!. We then define@5#
^Tmn&[C~u!diag~e,p,p sinh2h,p sinh2h sin2u!. ~2.33!
n--
III. RENORMALIZATION OF Tµn
In order to analyze the divergences of^Tmn& we make useof the adiabatic expansion of the modesgs , since we know@10# that ^Tmn& and ^Tmn&adiabatic have the same ultraviolebehavior. Up to second adiabatic order we have@10#
gsA5
1
A2Vs
expF2 i Eu
du8Vs~u8!G , ~3.1a!
ugsAu25
1
2Vs5
1
2vs1
vs
8vs4 2
3vs2
16vs5 1••• , ~3.1b!
ugsAu25
1
2VsS Vs
2
4Vs2 1Vs
2D 5vs
22
vs
8vs2 1
5vs2
16vs3 1••• ,
~3.1c!
~gsAgs
A* 1gsAgs
A* !521
2
Vs
Vs2 52
1
2
vs
vs2 1••• , ~3.1d!
wherevs2(u)5s21x(u).
We regularize our integrals by introducing a nocovariant cutoff L in physical momentum, which corresponds to a comoving momentum cutoffsm5C1/2(u)L. We
tht
blda
gaote
tio
drm
ga
qua-hal-
en
57 87EFFECTIVE EQUATION OF STATE FORA . . .
shall work with a fixed physical cutoffL, so that the comov-ing cutoff sm depends on the conformal timeu.
We have three physical parameters to renormalize:mass of the pionmp , the self-interaction coupling constanl, and the~dimensionless! coupling constant to gravityj,necessary in order for the field theory to be renormalizaeven in Ricci flat spacetimes, such as the one considerethis paper@11#. The quadratic divergences in the gap eqution will be removed by mass renormalization, and the lorithmic divergences will be subtracted by renormalizationthe coupling constantsl andj. In addition, we shall see thathere is an extra quartic divergence in both the energy dsity and the pressure, coming from the mode integrals^Tmn& that carry an extra factor ofk2, which can be removedby renormalizing the cosmological constant.
First we examine the divergences in the gap equa~2.28!. From the adiabatic expansion~3.1!, we know that themode integral appearing in this equation has a quadraticvergence. In order to remove this divergence, we perfomass renormalization by subtracting the regularizedequation for the vacuum state:
C~u!m2
l52 v 2~u!1C~u! f p
2 1N
2 E0
sm s2ds
2p2
1
~s21e2u!1/2.
~3.2!
e
e,in--f
n-in
n
i-
p
This yields for the gap equation
x~u!
l5
C~u!m2
l1r i
2~u!2C~u! f p2 1NE
0
sm s2ds
2p2
3F ~2ns11!ugs~u!u221
2~s21e2u!1/2G . ~3.3!
Note that the second integral isindependentof time. Oncewe have removed the quadratic divergences in the gap etion, we are only left with logarithmic divergences, whicare subtracted by the following coupling constant renormizations
1
l5
1
lR2
N
4 E0
sm s2ds
2p2
1
~s21e2u!3/2, ~3.4a!
S j21
6D 1
l5S jR2
1
6D 1
lR. ~3.4b!
Note thatj51/65jR is a fixed point of the renormalizationflow equations@12#.
The renormalized form of the gap equation is then givby the following:
ergencese
rede
x~u!
lR52 vR
22NC~u!m2
16p2 1r i21NE
0
sm s2ds
2p2 ~2ns11!ugs~u!u22NC~u!
2 E0
L k2dk
2p2
1
~k21m2!1/2
2N@ x~u!2C~u!m2#
4 E0
L k2dk
2p2
1
~k21m2!3/2. ~3.5!
We now proceed to write the regularized expressions for the energy and isotropic pressure and analyze the divappearing in the mode integrals. We shall takejR51/6 as suggested in@12#. The physics behind this choice is clear. If onconsiders an arbitrary composite scalar field~such as the meson field studied here! and an effective field theory at a scaleL,and one carries out a renormalization group analysis in the leading large-N approximation~or a fully improved one-looprenormalization group approximation!, j51/65jR is found to be an attractive renormalization group fixed point in the infralimit @12#. This means that even if there are corrections toj51/6 at large scales~of orderL!, the observed low energy valuof the couplingj tends to a physical value ofjR51/6, as one evolves into the infrared. The choicej51/65jR also impliesthat the divergences of both the energy and the pressure can be obtained at adiabatic order zero@10#.
Now we examine the divergences present in the mode integrals of the energy density and pressure. Recall that
E~u
in
pa
i-
mrge-o-yicic
enuu
,
es-
en-
sec-
the
,a-
88 57MELISSA A. LAMPERT AND CARMEN MOLINA-PARIS
For the pressure the contribution coming from the modetegrals is
N
6 E0
sm s2ds
2p2 ~2ns11!$ugsu21~s22x ! ugsu2%. ~3.10!
Again using the adiabatic mode expansion, the divergentof this integral is given by
C2~u!p0[N
6E
0
sm s4ds
2p2
1
As21x
5Nsm
4
48p2 2Nsm
2 x
48p2 1Nx 2
64p2 lnS 4sm2
xD
27Nx 2
384p2 1OS x 3
sm2 D . ~3.11!
Notice thate0Þ2p0 , as is required by general covarance~see Appendix C!. We must then enforce covariancebyhand. We mention here that the introduction of a momentucutoff does not spoil covariance at the level of the enedensity~see Appendix C!. On the other hand, we must carfully handle the subtractions that will lead to the finite istropic pressure@13#. The quartic subtraction in the energdensity and pressure is a renormalization of the cosmologconstant; in other words, a subtraction of the cosmologvacuum energyNL4/16p2.
If we now define
eR5e2Nsm
4
16p2C2~u!, ~3.12a!
pR5p2p02e01Nsm
4
16p2C2~u!, ~3.12b!
we can easily show that the energy density and pressurefinite, by making use of Eqs.~3.2! and ~3.4!.
In order to numerically evaluate the physical energy dsity and pressure, we must also make sure that the vacenergy that we are measuring with respect to is zero~i.e.,when we are at the minimum of the potential!. This vacuumenergy is calculated in the ‘‘out’’ regime of the collisionwhen ^s&5 f p , ^pW i&50, x5mp
2 , and the mode functionsare the zeroth order adiabatic ones withvs
25s21C(u)m2.The renormalized vacuum energy is given by
evac5m2f p
2
22
m2v2
22
m4
4l2
NL4
16p2
1N
2C2 E0
sm s2ds
2p2 As21m2C. ~3.13!
-
rt
y
alal
are
-m
The vacuum value of the pressure ispvac52evac. Thesevalues must be subtracted from Eqs.~3.12! so that when wehave reached the ‘‘out’’ regime, the energy density and prsure go to zero.
We calculate now the trace of the energy-momentum tsor, and show that these definitions ofeR andpR give a finitetrace. We have from its definition
2^Tmm&R53pR2eR . ~3.14!
If we make use of Eqs.~3.12!, we can write
2C2~u!^Tmm&R5 vR
2x~u!13 j ir i~u!1Nx 2~u!
32p2 ,
~3.15!
where we have defined
v 252m2C~u!
lR1 f p
2 C~u!2Nm2C~u!
16p2 1Nsm
2
8p2 [ vR21
Nsm2
8p2 .
~3.16!
The first two terms in Eq.~3.15! are the renormalizedclassical trace of the energy-momentum tensor, and theond term is the one-loop quantum trace anomaly.
IV. NUMERICAL RESULTS
A. Energy-momentum tensor conservation
The renormalized energy-momentum tensor obeysconservation law
¹m^Tmn&R50. ~4.1!
Using the Christoffel symbols tabulated in Appendix Awe find that then5u component of the conservation eqution takes the form
eR~u!13@pR~u!1eR~u!#50. ~4.2!
In terms of the variables
ER5E2Nsm
4
16p2 , ~4.3a!
PR5P2C2~u!~p01e0!1Nsm
4
16p2 , ~4.3b!
we can rewrite the conservation equation as
ER~u!13PR~u!2ER~u!50. ~4.4!
Recall that
ER51
2@ r i
21r i2~ x21!#2 j ir i2
x v 2
22
x 2
4l1
N
2 E0
sm s2ds
2p2 ~2ns11!$ugsu21~s21x !ugsu2%2Nsm
4
16p2 . ~4.5!
Therefore
57 89EFFECTIVE EQUATION OF STATE FORA . . .
ER5 r i r i1~ x21!r i r i11
2xr i
22x v 2
22x v v2
x x
2l2~ j ir i1 j r i !1
N
2 E0
sm s2ds
2p2 ~2ns11!
3$gsgs* 1gsgs* 1~s21x !~gsgs* 1gsgs* !1 xugsu2%1N
2
sm3
2p2 ~2nsm11!$ugsm
u21~sm2 1x !ugsm
u2%2Nsm
4
4p2 , ~4.6!
in
tio-
i
ite
lib-an
-‘ef-
os-thewill
theletly atWeered
ve-f
in toure
a
alof
te
ure
where we have taken into account the contribution comfrom the upper limit of the mode integral, sincesm depends
on u. Making use of the identitiesv i5 v i , j i53 j i , theequations of motion, and using the adiabatic approximafor gsm
~we can always choosesm to be a high enough comoving momentum so that the adiabatic approximationvalid in this limit!, we then obtain
ER52x v 223 j ir i1N
4p2 sm3 ~2nsm
11!vsm2
Nsm4
4p2 .
~4.7!
The occupation numbernsmgoes to zero for largesm , so we
can neglect it. Expanding outvsmyields
vsm5Asm
2 1x5smS 11x
2sm2 2
x 2
8sm4 1••• D , ~4.8!
so that we have
ER52x v 21Nxsm
2
8p2 23 j ir i2Nx 2
32p2 , ~4.9!
or equivalently
ER52x v R223 j ir i2
Nx 2
32p2 . ~4.10!
If we compare this expression with Eq.~3.15!, we can easilysee that the conservation equation is satisfied.
B. Effective equation of state
Now that we have finite equations for the energy densand pressure, we can evolve them in proper time and num
FIG. 1. Proper time evolution of thex field for the following
initial conditions. Solid line:s(t0)5sT , p i(t0)50, and s(t0)
50. Dashed line:s(t0)5sT , p i(t0)50, ands(t0)521.
g
n
s
yri-
cally investigate their behavior. Since we have a nonequirium situation, it does not really make sense to calculateactual equation of state, i.e.,p5p(e). However, sincee andp are both functions ofu, in regions where both are monotonically increasing or decreasing, one can determine an ‘fective’’ equation of statep5p(e). In this calculation, wedo not have two-body scattering, so that there are manycillations in the pressure. We expect at the next order in1/N expansion, when these effects are included that onebe able to extract an effective equation of state.
C. Numerical simulations
We choose the initial state at a temperature abovephase transition in thermal equilibrium, with all particmasses positive. The equations are solved self-consistenthe starting time to obtain the values of the mean fields.fixed the value ofx at the initial time as the solution of thgap equation in the initial thermal state. We also requithat the initial expectation values of thes andpW fields satisfypW 2(u0)1s2(u0)5sT
2 , wheresT is the thermal equilibriumvalue ofF at the initial temperatureT. The critical tempera-ture is Tc'160 MeV, so we chooseT5200 MeV, whichgivessT50.3 fm21. The mode functions are chosen to hatheir adiabatic values@see Eq.~2.24!#. We choose the coupling constantlR57.3 @2#, and take the conformal value othe coupling to gravityj5jR51/6 @12#.
In Fig. 1, we show the time evolution of thex field, whichis the effective mass squared for thef field. When this fieldbecomes negative, long-wavelength unstable modes beggrow exponentially. Therefore this field serves as a measof instability in the system. For initial conditions withnegative derivative, we find that thex field is negative forabout 3 fm/c of proper time. Figure 2 shows the numericcalculation of the energy density and pressure, in unitsfm24, for initial conditions where no instabilities arose. No
FIG. 2. Proper time evolution of the energy density and press
for the initial conditionss(t0)5sT , p i(t0)50, ands(t0)50.
ret flu
eye
nd
tun-f t
ll
gy
etutuy.tia
thd
chsitytatend
ckly,st-assativeion
-pe-
on-s
t-
u
tia
tial
90 57MELISSA A. LAMPERT AND CARMEN MOLINA-PARIS
that the system quickly reaches the ‘‘out’’ regime andlaxes to its vacuum values. Figure 3 shows the same ploinitial conditions with an instability. For the unstable initiaconditions more energy density was produced initially, bboth the energy density and the pressure drop off vquickly, as is expected~since the system is cooling verrapidly!. It is interesting to note that the pressure becomnegative whenx is negative. If we define the speed of souc0
2(t) by p5c02(t)e, we notice thatc0
2<1.Figure 4 shows the conservation of the energy-momen
tensor for stable initial conditions. Notice that it is not coserved for short proper times. Because of the presence oLandau pole it is not possible to take the physical cutoffLvery large~we use 800 MeV!. Because of this rather smacutoff, the occupation numbernsm
is not actually zero, whichit was assumed to be for the derivation of the enermomentum conservation@see the discussion after Eq.~4.6!#.As soon as the cutoffsm becomes large enough, then thoccupation number goes to zero and the energy-momentensor is conserved. Figure 5 shows the energy-momentensor conservation for initial conditions with an instabilitWe see a qualitatively similar behavior, regardless of iniconditions.
V. CONCLUSIONS
We have shown how to regularize and renormalizeenergy-momentum tensor for a spherically symmetric mo
FIG. 3. Proper time evolution of the energy density and press
for the initial conditionss(t0)5sT , p i(t0)50, ands(t0)521.
FIG. 4. Energy-momentum tensor conservation, for the ini
conditionss(t0)5sT , p i(t0)50, ands(t0)50.
-or
try
s
m
he
-
mm
l
eel
with a time-dependent comoving momentum cutoff whibreaks covariance. We computed the finite energy denand pressure, and numerically examined the equation of sand proved energy conservation for this system. We fouthat the energy density and pressure both decrease quiwhich is expected for a rapidly cooling system. An intereing feature of the expansion is that when the effective msquared is negative, the pressure is also negative. A negpressure implies cavitation, which could mean the formatof domains, i.e., DCCs.
ACKNOWLEDGMENTS
The authors would like to thank Yuval Kluger, Emil Mottola, and John Dawson for useful discussions; and give scial thanks to Fred Cooper and Salman Habib for their ctributions. The University of New Hamphshire wasupported by the U.S. Department of Energy~DE-FG02-88ER40410!. One of us~M.A.L.! would like to thank theLos Alamos National Laboratory for its hospitality.
APPENDIX A: CHRISTOFFEL SYMBOLS
The only non-vanishing Christoffel symbols for the meric
gmn5C~u!diag~1,21,2sinh2h,2sinh2h sin2u!,~A1!
with
C~u!5a2~t!5t25e2u/m2, ~A2!
are
Ghhu 51, Guu
u 5sinh2h,
Gwwu 5sinh2h sin2u, Guu
h 52sinhh coshh,
Gwwh 52sinhh coshh sin2u, Gww
u 52sinu cosu,
Guuu 5Ghu
h 5Guuu 5Gwu
w 51,
Guhu 5Gwh
w 5cothh,
Gwuw 5cotu.
re
l
FIG. 5. Energy-momentum tensor conservation, for the ini
conditionss(t0)5sT , p i(t0)50, ands(t0)521.
va
m
he
m
-a
inia
nheak
nac
int-
ions.
57 91EFFECTIVE EQUATION OF STATE FORA . . .
The Christoffel symbols are used to derive the consertion law for the energy-momentum tensor.
APPENDIX B: ENERGY-MOMENTUM TENSORFOR A PERFECT FLUID
For a perfect fluid we can write the energy-momentutensor as follows:
Tmn52pgmn1~e1p!umun , ~B1!
whereum are the components of the velocity vector of tfluid, such that it is normalized, i.e.,gmnumun51, which cor-responds to a timelike vector field. In the reference frawhere the fluid is at rest we can write
um5~uu ,uh ,uu ,uw!5~uu ,0W !, ~B2!
and the normalization condition simply implies thatguu5uuuu5C(u). It is then straightforward to obtain the components of the energy-momentum tensor in this coordinsystem
Tuu5guue5C~u!e,
Thh52ghhp5C~u!p,
Tuu52guup5C~u!p sinh2h,
Tww52gwwp5C~u!p sinh2h sin2u. ~B3!
Then we can write
^Tmn&[C~u!diag~e,p,p sinh2h,p sinh2h sin2u!.~B4!
APPENDIX C: REGULARIZATION BY COVARIANTGEODESIC POINT-SPLITTING
The regularization scheme~introducing a physical mo-mentum cutoffL! used to render the mode integrals finitethe energy density and isotropic pressure is not a covarmethod. Since we must obtain covariant results~we want toobtain ^Tmn& renormalized), we have to proceed with care, iorder to perform the physical subtractions that will yield tphysical finite energy density and pressure. We shall muse of the results obtained by Christensen@4#, in order toanalyze the non-covariant structure on the divergences ie0and p0 , since we know that covariant point-splitting inspatial direction is equivalent to regularization by introduing a momentum cutoff.
Christensen@4# obtained
^Tmn&quartic5 limx8→x
1
2p2
1
~slsl!2 S gmn24smsn
~slsl! D ,
^Tmn&quadratic52 limx8→x
1
4p2
1
~slsl!
x
2 S gmn22smsn
~slsl! D ,
^Tmn& logarithmic52 limx8→x
1
4p2
x4
8gmnS g1
1
2lnU xs
2 U D .
~C1!
-
e
te
nt
e
-
If we choosesm to be a spacelike vector such that
s is j51
3gi j ~slsl! and su50 with i , j 5h,u,w,
~C2!
we can write
^Tuu&quartic5 limx8→x
1
2p2
1
~slsl!2 C~u!,
^Thh&quartic5 limx8→x
1
2p2
1
~slsl!2
C~u!
3,
^Tuu&quadratic52 limx8→x
1
4p2
1
~slsl!
x
2C~u!,
^Thh&quadratic5 limx8→x
1
4p2
1
~slsl!
x
2
C~u!
3,
^Tuu& logarithmic52 limx8→x
1
4p2
x4
8C~u!S g1
1
2lnU xs
2 U D ,
^Thh& logarithmic5 limx8→x
1
4p2
x4
8C~u!S g1
1
2lnU xs
2 U D .
~C3!
From Appendix B we know thatTuu5C(u)e and thatThh5C(u)p and from the previous equations~C3! we canmake the following identifications:
equartic5 limx8→x
1
2p2
1
~slsl!2 ,
pquartic5 limx8→x
1
2p2
1
~slsl!2
1
3,
equadratic52 limx8→x
1
4p2
1
~slsl!
x
2,
pquadratic5 limx8→x
1
4p2
1
~slsl!
x
2
1
3,
e logarithmic52 limx8→x
1
4p2
x4
8 S g11
2lnU xs
2 U D ,
plogarithmic5 limx8→x
1
4p2
x4
8 S g11
2lnU xs
2 U D . ~C4!
Thus it follows
equadratic53pquadratic,
equartic523pquartic ,
e logarithmic52plogarithmic . ~C5!
We can now compare these general results for spatial posplitting with Eqs. ~3.9! and ~3.11! by making use of thenatural identificationslsl}L22. It is easy to see that thequartic, quadratic and logarithmic terms of these expressfulfill the requirements of Eq.~C5!, as we wanted to show
siro
y,
. D
,.
nd
D
92 57MELISSA A. LAMPERT AND CARMEN MOLINA-PARIS
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